Jinsheng Ning

Comments on Two Dimensional Convolutions of the Geodetic Problems in Planar and Spherical Coordinates

Almost all integral formulae in geodetic problems can be expressed in a convolution form, which makes it possible to perform a convolution evaluation by the fast Fourier transform or the fast Hartley one. The 2D convolution forms are usually in planar or spherical coordinates. Unfortunately, the accuracy of results from the evaluation of the 2D convolution in planar coordinates is lower than that in spherical coordinates and the numerical integration in the past. This conflict is caused by inadequately ignoring terms of the kernel function which follow the principal term. To satisfy the convolution theorem, it is necessary for the 2D convolutions in spherical coordinates to take the latitude approximation in their kernel function. Consequently, it leads to larger errors in results. The proposed new idea is that all terms of Stokes kernel functions can be taken into account by transforming all variables sin(ψ/2) of the functions to the l, a straight line length corresponding to the spherical distance ψ, and the functions could strictly be expressed in planar coordinates. In this case, any approximation will be avoided. Therefore, the results from the 2D convolution in planar coordinates can be obtained with much better accuracy than that in spherical coordinates, and they are very close to that from the 1D convolution or the numerical integration. Based on the above discussion, the evaluations of the Stokes formula are carried out using the 1D convolution and the 2D convolution in planar coordinates, and also that in spherical coordinates for the comparisons between them.

A multiscaling-based semi-analytic orbit propagation method for the catalogue maintenance of space debris

ABSTRACT The choice of orbit propagation method is essential for orbit prediction (OP) and determination (OD) of space debris, requiring both high accuracy and computational efficiency. This paper presents a semi-analytic method using the multiscaling technique. The 7-day OP errors are less than 200 m for orbits above 800 km. The 5-year semi-analytic solutions are well fitted to the numerically propagated orbit. OD performance of the semi-analytic method is examined using real data, and the determined position accuracy is at dozens of metres. The computational efficiency of the semi-analytic method against the numerical method is improved by about 95 percent.

Applications of the Fictitious Compress Recovery Approach in Physical Geodesy

The fictitious compress recovery approach is introduced, which could be applied to the establishment of the Runge-Krarup theorem, the determination of the Bjerhammar’s fictitious gravity anomaly, the solution of the “downward continuation” problem of the gravity field, the confirmation of the convergence of the spherical harmonic expansion series of the Earth’s potential field, and the gravity field determination in three cases: gravitational potential case, gravitation case, and gravitational gradient case. Several tests using simulation experiments show that the fictitious compress recovery approach shows promise in physical geodesy applications.

Study on recovering the Earth’s potential field based on GOCE gradiometry

Given the second radial derivative Vrr(P)|∂S of the Earth’s gravitational potential V(P) on the surface ∂S corresponding to the satellite altitude, by using the fictitious compress recovery method, a fictitious regular harmonic field rrVrr(P)* and a fictitious second radial gradient field Vrr*(P) in the domain outside an inner sphere Ki can be determined, which coincides with the real field Vrr(P) in the domain outside the Earth. Vrr*(P) could be further expressed as a uniformly convergent expansion series in the domain outside the inner sphere, because rrVrr(P)* could be expressed as a uniformly convergent spherical harmonic expansion series due to its regularity and harmony in that domain. In another aspect, the fictitious field V*(P) defined in the domain outside the inner sphere, which coincides with the real field V(P) in the domain outside the Earth, could be also expressed as a spherical harmonic expansion series. Then, the harmonic coefficients contained in the series expressing V*(P) can be determined, and consequently the real field V(P) is recovered. Preliminary simulation calculations show that the second radial gradient field Vrr(P) could be recovered based only on the second radial derivative Vrr(P)|∂S given on the satellite boundary. Concerning the final recovery of the potential field V(P) based only on the boundary value Vrr(P)|∂S, the simulation tests are still in process.